
days. Evaluating both sides of (11) at instances t
i
, i = 1, 2, . . . , N, we will have
F (α
1
(t
1
) , α
2
(t
1
) , . . . , α
n
(t
1
) ; t
1
) = S (t
1
) ,
F (α
1
(t
2
) , α
2
(t
2
) , . . . , α
n
(t
2
) ; t
2
) = S (t
2
) ,
.
.
.
F (α
1
(t
N
) , α
2
(t
N
) , . . . , α
n
(t
N
) ; t
N
) = S (t
N
) .
(12)
Note that (12) is linear with respect to each α
k
(t
i
), k = 1, 2, . . . , n, i = 1, 2, . . . , N. Nonetheless,
(12) contains N equations with respect to n · N unknowns. At this, N ≫ n and the system
is under-determined. Therefore, direct methods can not be applied for solving (12). On the
other hand, in order to determined unknowns α
k
(t
i
), k = 1, 2, . . . , n, i = 1, 2, . . . , N, we can
apply either efficient numerical methods of linear programming, choosing, e.g., P
BTC
as a cost
function, or we can solve the equivalent problem of numerical minimization
||F (α
1
(t
i
) , α
2
(t
i
) , . . . , α
n
(t
i
) ; t
i
) − S (t
i
)|| → min
0≤α
k
(t
i
)≤1
, k = 1, 2, . . . , n i = 1, 2, . . . , N.
Here, ||·|| denotes an appropriate norm. In this case, it can be the l
2
-norm. Then, the problem
formulation will be: determine the solution to the following numerical minimization problem:
|F (α
1
(t
i
) , α
2
(t
i
) , . . . , α
n
(t
i
) ; t
i
) − S (t
i
)|
2
→ min
0≤α
k
(t
i
)≤1
, k = 1, 2, . . . , n i = 1, 2, . . . , N.
The Case of Single Alpha (α)
In the case of single absorption, consideration of particular models leads to the following ex-
pression for Π
BTC
:
Π
BTC
=
∂
∂t
"
ln
n
X
k=1
α · P
k
· R
k
!
+ ln
1
n
n
X
j=1
T
′
j
!
− ln b − ln h + ln d
#
=
=
∂
∂t
"
ln
α ·
n
X
k=1
P
k
· R
k
!
+ ln
1
n
n
X
j=1
T
′
j
!
− ln b − ln h + ln d
#
=
=
∂
∂t
"
ln α + ln
n
X
k=1
P
k
· R
k
!
+ ln
1
n
n
X
j=1
T
′
j
!
− ln b − ln h + ln d
#
.
Similarly, in this case, we have
ln P
BTC
= ln α + ln
n
X
k=1
P
k
· R
k
!
+ ln
1
m
m
X
j=1
T
′
j
!
− ln b − ln h + ln d.